The laws of logarithms

Revision – true or false?

Recap

log₂ 4 = 2

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log₂ 16 = 4

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log₂ 64 = 6

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log₂ 4 + log₂ 16 = 6

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Evaluate

log₃ 3 = 1

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log₃ 27 = 3

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log₃ 81 = 4

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log₃ 3 + log₃ 27 = 4

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log₁₀ 100 = 2

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log₁₀ 1000 = 3

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log₁₀ 100000 = 5

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log₁₀ 100 + log₁₀ 1000 = 5

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What’s going on?

log₂ 4 + log₂ 16 = log₂ 64

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log₃ 3 + log₃ 27 = log₃ 81

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log₁₀ 100 + log₁₀ 1000 = log₁₀ 100000

log₂ 2 + log₂ 32 = log₂ 64

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log₃ 9 + log₃ 9 = log₃ 81

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log₁₀ 2 + log₁₀ 50000 = log₁₀ 100000

Find the value so that the result is still true.

for logarithms of the same base

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1^st law of logarithms

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log a + log b = log ab

Can you arrange the cards to give a convincing proof? You may need to include some extra algebraic steps or explanations if you think they would help to make the argument clearer or more convincing.

log₂ 4 = 2

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log₂ 32 = 5

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log₂ 8 = 3

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log₂ 32 - log₂ 4 = 3

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Evaluate

log₃ 3 = 1

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log₃ 9 = 2

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log₃ 27 = 3

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log₃ 27 - log₃ 9 = 1

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log₁₀ 100 = 2

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log₁₀ 1000 = 3

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log₁₀ 0.1 = -1

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log₁₀ 100 - log₁₀ 1000 = -1

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What’s going on?

log₂ 32 - log₂ 2 = log₂ 16

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log₃ 90 - log₃ 3 = log₃ 30

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log₁₀ 20 - log₁₀ 4 = log₁₀ 5

Find values so that the result is still true.

Find a different set of values so that the result is still true.

for logarithms of the same base

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2^nd law of logarithms

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log a - log b = log ^a/_b

Rearrange the 6 lines to prove the 2^nd law

3log₂ 4 = 6

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log₂ 64 = 6

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Evaluate

2log₃ 3 = 2

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log₃ 9 = 2

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4log₁₀ 10 = 4

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log₁₀ 10000 = 4

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Write a law that is suggested by these calculations.

for logarithms of the same base

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3^rd law of logarithms

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n log a = log aⁿ

Rearrange the lines to prove the 3^rd law

Justify the following results

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log_a a = 1

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log_a 1 = 0

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log_a aⁿ = n

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Watch the video

Express in terms of log p, log q, and log r

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1. log pq 2. log pqr 3. log p/q

4. log pq/r 5. log p/qr 6. log p²q

7. log q/r² 8. log p√q 9. log p²q³/r

10. log √(q/r) 11. log qⁿ 12. log pⁿq^m

13. log 2pq 14. log ½ pq 15. log 2pq²

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Simplify

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16. log p + log q 17. 2 log p + log q

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18. log q – log r 19. 3 log q + 4 log p

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20. n log p – log q 21. log p + 2log q - 3log r

22. log p – log 2 23. 2 log p – p log 2

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24. log p + log q – log 3 25. 3log p – ½ (log q + log r)

Match up the equivalent expressions

Find the odd one out in each row.

Write a logarithmic statement equivalent to the odd one out.

Some exam question

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Extension

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an O-level question from 1960:

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